Papers

Modern conditional generative models face significant challenges when learning complex covariate dependencies. While sufficient dimension reduction (SDR) provides a principled approach to compress these dependencies, traditional SDR frameworks were not formulated for conditional generation. To bridge this gap, we propose Belted Engression, a unified and architecturally parameter-efficient framework for generative distributional regression. Our approach establishes an end-to-end compress-then-generate paradigm driven by sufficient representation learning, embedding a structural bottleneck into the generative architecture. Theoretically, we prove that the standard SDR condition is equivalent to a law-preserving generative factorization, which is achieved at the global optimum of the population Belted Engression objective. Furthermore, by uncovering a localized Bernstein-type control for the energy-score loss, we establish finite-sample convergence rates that are sharper than those of existing results. We also prove that this belted architecture is strictly smaller, operating with an asymptotically vanishing parameter count relative to the unstructured baseline. Extensive simulations and real-world applications demonstrate that Belted Engression achieves superior distributional prediction and SDR recovery with fewer trainable parameters.

A recent pre-print [arXiv:2609.19214] proposed a three-dimensional (3D) strongly aperiodic monotile: a shape that tiles Euclidean space only non-periodically and which admits no symmetry of infinite order. The proof takes the 3D Chair tile identified previously by Lee and Moody, and adds geometric decorations to the faces so as to force non-periodicity (without these decorations The Chair also admits periodic tilings). Here we establish general requirements on face decorations to achieve the same end, in order to facilitate the search for physical realisations. We find that the requirements are minimal. We provide matching rules using three colours of arrow that are equivalent to the original rules, in that they force the same local and global configurations. These rules force Chairs to compose into `Superchairs' with doubled linear dimensions. In this process the matching rules themselves compose uniquely. We find that the same global structure can be forced using simpler rules based on the colours of squares, regardless of orientation. Any physical system encoding these rules (geometrically or otherwise) will force the same strongly aperiodic monotilings. We provide simple examples.

Let $\sigma_{n,d}$ denote the elementary symmetric Boolean function of $n$ variables and degree $d$. We completely determine the sensitivity, average sensitivity, and block sensitivity of $\sigma_{n,d}$ for every $1\le d\le n$. Using Lucas' theorem, we obtain a uniform binary description of the Hamming-weight value sequence, from which the sensitivity and average-sensitivity formulas follow and the computation of block sensitivity reduces to at most four explicit candidates. Combining these results with the arbitrary-degree formula for certificate complexity, we determine the exact relations among sensitivity, block sensitivity, and certificate complexity. We also prove a general result for symmetric Boolean functions: every nonconstant symmetric Boolean function $f$ satisfies \[ \bs(f)\le \max\{s(f),C(f)-1\}. \] Consequently, only the three patterns \[ s=\bs=C,\qquad s=\bs<C,\qquad s<\bs<C \] can occur for nonconstant symmetric Boolean functions. For elementary symmetric Boolean functions, we give necessary and sufficient conditions for each of these three patterns, thereby completely classifying the relations among $s(\sigma_{n,d})$, $\bs(\sigma_{n,d})$, and $C(\sigma_{n,d})$. In particular, we obtain a necessary and sufficient characterization of the full strict hierarchy \[ s(\sigma_{n,d})<\bs(\sigma_{n,d})<C(\sigma_{n,d}), \] and exhibit infinite families for which it holds.

For integers $r\ge2$ and $n\ge1$, let $$ F_{r,n}(q)=\prod_{k=1}^{n}\frac{1-q^{rk}}{1-q^k}. $$ The coefficient of $q^j$ in \(F_{r,n}(q)\) counts partitions of $j$ into parts at most $n$, each occurring at most $r-1$ times. Hughes proved that $F_{2,n}(q)=\prod_{k=1}^{n}(1+q^k)$ is unimodal for every $n\ge 1$. This result was reproved by Stanley using an algebraic approach and Odlyzko and Richmond using an analytic approach. Almkvist conjectured that $F_{r,n}(q)$ is unimodal in the following two cases: every even $r$ and every $n\ge 1$; every odd $r$ and every $n\ge 11$. He proved that this conjecture is true for $3\le r \le 20$ and $r=100,101$. In this paper, we completely settle the conjecture.

We show that Arc Kayles is PSPACE-complete. This solves a question raised by Schaefer in 1978.

We classify all smooth local conservation laws of three-dimensional homogeneous incompressible Euler flow in velocity and pressure, allowing explicit time and position and derivatives of arbitrary finite order. Modulo currents vanishing on solutions and currents with identically zero divergence, the classical mass, momentum, angular-momentum, kinetic-energy and helicity balances, together with generalised time-dependent momentum, exhaust the classification. We also classify all smooth finite-order local symmetry characteristics and adjoint symmetries: differential expressions satisfying the linearised Euler equations and their formal adjoint on solutions, respectively. Every symmetry characteristic is equivalent on solutions to one induced by a point transformation. Every adjoint symmetry has a multiplier representative, whose product with the Euler equations is identically a divergence. These representatives exhaust the restrictions of all local conservation-law multipliers to solutions. The conservation-law proof has three stages: classification of adjoint symmetries using their determining equations, verification of explicit multiplier identities, and the correspondence between multiplier restrictions and current classes. A common analysis of highest-derivative responses reduces both symmetry problems to order at most one.

We prove uniqueness, up to phase rotations and translations, of the ground state for the $L^2$-critical boson star equation \[ (\sqrt{-\Delta}+1)Q =\bigl(|x|^{-1}* |Q|^2\bigr)Q, \qquad x\in\mathbb R^3. \] We also determine the kernels of the real and imaginary linearized operators. As a consequence, we show the Lieb-Yau uniqueness conjecture holds when the mass is sufficiently close to the Chandrasekhar mass.

We consider the two-dimensional barotropic compressible nematic liquid crystal flows with a planar director in the whole space or in the half-space. Under the assumptions that the shear viscosity is a positive constant and that the bulk viscosity is given by $\lambda(\rho)=\rho^\beta$ with $\beta>4/3$, we establish the global existence and uniqueness of strong solutions. It should be mentioned that our results are obtained without any restrictions on the size of the initial data and allow for the presence of vacuum. In particular, the initial orientation field is not required to satisfy any geometric angle condition.

We study optimal stopping under dynamic risk measures with simultaneous ambiguity in the probability model and the discount rate. We introduce a paired ambiguity framework combining Girsanov model uncertainty with cash subadditive risk evaluation and characterize the stopping value by an upper reflected backward stochastic differential equation (BSDE). We establish structural properties of the resulting stopping operator and study quadratic drivers associated with entropic risk measures, obtaining explicit stopping rules in several benchmark cases. We then develop a deep learning scheme for the reflected quadratic BSDE. The convergence analysis uses discrete reflection and truncation to reduce the quadratic problem to a globally Lipschitz system and combines reflected BSDE discretization estimates with neural network approximation errors. Numerical experiments for American options illustrate the effects of discount rate and entropic ambiguity on stopping values and exercise decisions.

In this paper, we investigate two-dimensional compressible nematic liquid crystal flows with a planar director. We assume that the shear viscosity is a positive constant and that the bulk viscosity is given by $\lambda(\rho)=\rho^\beta$ with $\beta>4/3$. In the periodic domain or in a bounded simply connected smooth domain with Navier-slip and Neumann boundary conditions, we establish the global existence and uniqueness of strong solutions without any restrictions on the size of the initial data. In particular, no geometric angle condition is imposed on the initial orientation field. Moreover, we derive a time-uniform upper bound for the density and establish exponential decay estimates for the strong solutions.

For a complex $\varepsilon$-log canonical Fano surface of Picard number one, we prove that the anticanonical volume is at least $\varepsilon^2/100$ and that every integral nef and big Weil divisor has volume at least $\varepsilon^7/1728$. Both exponents are optimal. For $\varepsilon$-log canonical log Calabi–Yau surface pairs of arbitrary Picard number, we prove a lower bound of order $\varepsilon^9/(1+\log_2(1/\varepsilon))$, without a bigness assumption or further restrictions on the boundary coefficients. We also prove $\tau(X,H)\le18\varepsilon^{-3}$ for the pseudo-effective threshold on every projective $\varepsilon$-log canonical surface, with optimal exponent three. A volume-to-birationality estimate gives a sufficient bound of order $\varepsilon^{-11/2}\sqrt{1+\log_2(1/\varepsilon)}$ in the log Calabi–Yau setting.

We show that for closed negatively curved Riemannian manifolds whose length spectrum is exponentially separated, the length spectrum is a local isometry invariant. In particular, this holds for a dense set of negatively curved metrics. The main tool used in the proof is a finitary version of the Livshits theorem with an exponential tail.

This paper studies the two-dimensional nonhomogeneous incompressible nematic liquid crystal flows with planar orientation fields taking values in $\mathbb{S}^1$. For the initial-boundary value problem in bounded domains with density-dependent viscosity, we establish the global existence and exponential decay of strong solutions with initial density allowing vacuum, provided that $\|\nabla \mu(\rho_0)\|_{L^q}$ is sufficiently small for some $q>2$. The smallness condition is automatically satisfied when $\mu$ is constant, and hence the result yields the global existence of strong solutions for arbitrarily large initial data in the constant viscosity case. Furthermore, for the Cauchy problem with constant viscosity and either vacuum or non-vacuum far-field density, we establish the global existence and large-time decay rates of strong solutions for arbitrarily large initial data. These results are obtained without imposing any geometric angle conditions on the initial orientation field.

Lagrangian duality is fundamental to optimization, from the development of solver techniques to the interpretation of solutions; yet its geometric intuition can be difficult to grasp through mathematical derivation alone. This tutorial explains duality and its implications in a highly visual way for a broad audience of students and practitioners. It develops the motivation for the Lagrangian, the two-player game interpretation, and the connection to Fenchel duality. Throughout, it emphasizes the geometric meaning of the solution from both the primal and dual

We develop a theory of equivariant local systems which categorifies genuine equivariant cohomology theories – such as Atiyah–Segal $K$-theory, Lurie's tempered cohomology, and the equivariant elliptic cohomology of Grojnowski, Greenlees, and Gepner–Meier – analogously to how the category of ordinary local systems categorifies singular cohomology. More precisely, we introduce, for a global space $X$ and a coefficient system $\mathscr{A} \colon \mathrm{Orb}^{\mathrm{op}} \to \mathrm{Pr}^{\mathrm{L}}$, the categories $\mathrm{LS}^{\mathrm{glo}}(X,\mathscr{A})$ and $\mathrm{LS}^{\mathrm{gen}}(X,\mathscr{A})$ of globally equivariant and genuine local systems, the latter generalizing the genuine stable category $\mathrm{Sp}^G$ of a compact Lie group $G$ to non-constant coefficients. When the coefficients come from an oriented abelian group stack $A$ over a locally complex periodic base, we also construct the category $\mathrm{LS}^\mathrm{temp}(X,A)$ of tempered local systems, extending Lurie's theory beyond finite groups; the defining condition is justified by a tempered form of the Atiyah–Segal completion theorem. We show that global sections induce an equivalence $\mathrm{LS}^{\mathrm{temp}}(BU(1),A) \simeq \mathrm{QCoh}(A)$ and we prove that $\mathrm{LS}^\mathrm{temp}(X,A)$ is a smashing localization of $\mathrm{LS}^\mathrm{gen}(X,A)$ if $X$ is an orbispace.

Convex spiral equicoverings were recently introduced by Espinosa-García, Martínez-Sandoval and Roldán-Pensado. They left open the question of whether every planar mass admits a convex $(3k,k+1)$-spiral equicovering. In this paper we give an affirmative answer to this question. To be precise, we prove the following: Given an integer $k \ge 2$, for every planar mass there is a fan consisting of $3k$ equal-mass sectors such that the union of every $k+1$ consecutive sectors is convex.

On the real line, Bárány, Kolossváry, and Troscheit [5] made exponential separation of analytic self-conformal systems checkable through a dual iterated function system on a space of analytic functions. The quantity their condition separates is the pre-Schwarzian derivative, and this observation lets us carry the construction to conformal iterated function systems in every dimension, where the pre-Schwarzian is $T_f=\nabla\log\|Df\|$. A pointwise separation of the pre-Schwarzian cocycle implies the strong exponential separation condition, and in dimensions one and two separation of the Schwarzian cocycle implies the same condition modulo Möbius maps. Each hypothesis is equivalent to a uniform gap condition, and these are $\mathcal{C}^2$-open. On the line they are also dense, so both separation conditions are $\mathcal{C}^2$-generic there; this sharpens the genericity theorem of Bárány, Kolossváry, and Troscheit, whose open and dense set carries only the plain condition. In the plane they are dense on Jordan domains with simply connected extension domains, and both separation conditions are $\mathcal{C}^2$-generic there too. In dimensions at least three every conformal map is Möbius, so no system satisfies the condition modulo Möbius maps, and the pre-Schwarzian hypothesis becomes a pole-separation condition, dense when the generators contract strongly enough and stay away from the similarities, but not dense in general. On Jordan domains, the planar genericity and a theorem of Feng and Rapaport [13] settle the dimension drop conjecture for $\mathcal{C}^2$-generic planar systems with injective generators.

We study the full angular dependence of the linearized Schrödinger form of the Calderón problem on the unit disk. We obtain an explicit continuous spectral diagonalization of its normal operator and hence a generalized singular system for the linearized forward map. To our knowledge, this is the first such diagonalization for the full angularly dependent, multiplicity-normalized linearized measurement map on the disk.

We study a contextual distributionally robust chance-constrained programming (C-DRCCP) model that controls violation probabilities conditional on a neighborhood of a target context. The ambiguity set is defined over the joint distribution of contexts and outcomes using the Wasserstein distance. For affine safety systems with right-hand-side uncertainty, we derive an exact mixed-integer programming (MIP) reformulation. The reformulation reveals that worst-case conditional risk is determined by two coupled uses of the transportation budget: moving empirical mass into the conditioning neighborhood and moving conditioned mass into the failure region. Equivalently, robust feasibility holds when the minimum transportation cost needed to reach the conditional risk boundary is at least the available budget. Calibrating the ambiguity set gives a finite-sample guarantee for the true conditional violation probability. This characterization leads to strong valid inequalities that exploit the contextual transport. Each contextual mass allocation determines both the conditional scenario weights and the budget available to move outcomes into failure, whereas conventional quantile cuts utilize only the induced weights. Accounting for the remaining budget generates strengthened quantile cuts that strictly dominate standard quantile bounds and yield tighter MIP coefficients. We also derive rank inequalities from an exchange property of contextual transportation costs, linking feasible failure patterns across contextual allocations. Extensive numerical experiments demonstrate that these inequalities close much of the linear-relaxation gap, speed up the solution process, and enable large-sample instances that are intractable for the baseline formulation.

In this contribution, a fluid flow problem on a general deforming domain for a Newtonian fluid in two and three space dimensions with artificial compressibility approximation is studied. We prove an estimate on the integral equicontinuity in time of the weak solutions under suitable domain regularity assumptions, which is independent on the compressibility parameter and serves as an alternative compactness argument for the convergence of weak solution sequences for vanishing compressibility. The corresponding estimate is obtained by remapping the problem onto a fixed reference domain and using appropriate divergence-preserving testfunctions involving the difference of two solutions at different points in time, thus defined with respect to different domains/coordinates.

We establish global well-posedness and a small-noise large deviation principle for the two-dimensional Dean-Kawasaki equation with Coulomb interaction $\partial_t\rho^\varepsilon=\Delta\rho^\varepsilon-\nabla\cdot(\rho^\varepsilon(V*\rho^\varepsilon))-\sqrt{\varepsilon}\nabla\cdot(\sqrt{\rho^\varepsilon}\circ\xi^{K(\varepsilon)})$, where $V=\lambda_V(\cos(\alpha_V)\nabla G+\sin(\alpha_V)J\nabla G)$. Here $\lambda_V\geq0$ is the interaction strength, $\alpha_V$ is the interaction angle, $G$ is the mean-zero Green function of $-\Delta$ on $\mathbb{T}^2$, $J$ is rotation by $\pi/2$, and $\xi^K$ is a finite-mode approximation of space-time white noise. For finite-entropy initial data of mass $M$ satisfying $\lambda_V\max\{\cos(\alpha_V),0\}M<8\pi$, the equation with the exact square-root coefficient is globally pathwise well posed at every finite Fourier cutoff in the class of stochastic kinetic solutions. Following Fehrman and Gess (arXiv:1910.11860), we prove a large deviation principle on $L^1((0,T)\times\mathbb{T}^2)$ under the scaling $K(\varepsilon)\to\infty$ and $\varepsilon K(\varepsilon)^4\to0$, with good rate function given by the quadratic control problem for the skeleton equation.

The unitary group of a von Neumann algebra on an $\aleph_0$-dimensional Hilbert space is polynomially weak$^*$-dense (equivalently, $G_{delta}$ dense or residual) in the normal unit ball if and only if the canonical finite atomic quotient of the von Neumann algebra is trivial. The analogous result holds for the usual weak$^*$ topology, in which case one can drop the normality constraint. In the latter form, this answers a question of T. Eisner and generalizes the counterpart for full bounded-operator algebras.

We study the relationship between the integrals of Yuan's concave transforms and the characteristic volumes of adelic line bundles on projective varieties. We establish an identity in dimension one and an inequality in higher dimensions. These results yield counterexamples to an identity conjectured by Yuan.

The local–global principle, which concerns the relationship between local structure and global parameters, has attracted considerable attention in extremal combinatorics over the past few decades. In this paper, we study how global density forces small dense subhypergraphs in uniform hypergraphs. For fixed $r\ge3$ and $s>1$, let $t_r(n,d,s)$ be the smallest integer $t$ such that every $n$-vertex $r$-graph of average degree at least $d$ contains a nonempty subhypergraph on at most $t$ vertices with average degree at least $s$. We show that the behavior of $t_r(n,d,s)$ undergoes a phase transition at $s=mainr/(r-1)$. We determine $t_r(n,d,s)$ and obtain asymptotically sharp bounds in several parameter regimes. This answers, up to polylogarithmic factors, a question of Feige and Wagner that was later restated as Problem~3.3 by Janzer, Sudakov and Tomon. In particular, when $r=3$ and $s=2$, our result implies a conjecture of Feige.

In this paper, we investigate the dynamics and long-time stability of hierarchical configurations in an $N$-point vortex system on $\mathbb{R}^2$ and in simply connected domains, without the Cantor-set restriction inherent in KAM theory. Since the Hamiltonian perturbation lacks a standard power-series expansion in the usual normalized variables, we develop a modified Birkhoff normal form based on successive ratios of adjacent variables, tailored to the hierarchical structure. A key feature is the preservation of the resulting natural exponent structure throughout the normal form procedure. Consequently, for $n$ sufficiently large, the hierarchical configuration persists for times of order at least $\varepsilon^{-\left(n-2(N-1)\right)}$.

We study automorphic transcendence degree for quotients of multivariate skew polynomial rings over a division ring D. We identify this invariant with relative growth dimension, the degree of the relative Hilbert polynomial, and a coordinate dimension determined by leading monomials. We establish invariance under finite module extensions, integral extensions, and investigate the Gelfand-Kirillov dimension under a local finiteness condition. When the coefficient automorphisms are independent modulo inner automorphisms, we prove that a quotient is automorphically normalizable precisely when its automorphic transcendence degree equals the number of nonnilpotent coordinate variables, and classify all such normalization subrings.

Prime and Cohen-Macaulay ideals of 2-minors of matrices of variables with linear resolution are classified.

The Sylvester simplex $\mathsf{Sylv}_d^k$ is a $d$-dimensional lattice simplex with exactly $k$ interior lattice points. Sylvester simplices are conjectured to be the volume maximizers among all $d$-dimensional lattice polytopes with exactly $k$ interior lattice points for any $k\geq 1$. Even stronger, it is conjectured that they maximize (entry-wise) the $h^\ast$-vector among all $d$-dimensional lattice polytopes with exactly $k$ interior lattice points. Yet, Sylvester simplices seem to be rarely studied in their own right. In particular, their Ehrhart-theoretic properties are far from being well understood. In the present article, we tackle this problem. We describe flag, regular and unimodular triangulations for the Sylvester simplices, and prove that their $h^\ast$-vectors are unimodal. Moreover, we explicitly determine the values of some entries of their $f^\ast$-vectors, and prove that they are Ehrhart magic positive up to dimension $6$ but not in dimension $7$. We conclude by detailing tables of Ehrhart-theoretic quantities (numbers of lattice points, Ehrhart polynomials, local and boundary $h^\ast$-vectors, $f^\ast$-vectors) for Sylvester simplices of dimensions 7 and lower.

We prove quantitative propagation of chaos in total variation for the unregularized repulsive Vlasov–Riesz–Fokker–Planck system on $\mathbb{T}^d$ and $\mathbb{R}^d$, $d\ge2$. For product initial data and potentials with singularity $|x|^{-s}$, $0<s<d/2$, we obtain an algebraic convergence rate for the $k$-particle marginals on a time interval independent of the particle number. The constants grow at most exponentially in $k$. The three-dimensional Coulomb case gives every rate $N^{-b}$ with $0<b<1/3$. The assumptions on its limiting solution hold for weighted Sobolev data with a Gaussian velocity moment and, on $\mathbb{R}^3$, a polynomial spatial moment. The proof combines conditioning of the initial product law, auxiliary particle equations, and weighted BBGKY estimates. An $L^1$ estimate and pointwise exponential energy bounds control the remainder flux in weighted $L^q$. For correlated initial data satisfying pointwise energy and marginal error bounds, the comparison extends to $0<s<d-1$.

Let $S_k(G)$ denote the sum of the $k$ largest eigenvalues of a graph $G$. Motivated by the classical Hoffman program for the spectral radius of a graph, we investigate an additive Hoffman-type problem for $S_k(G)$. For each fixed $k\geq 2$ and sufficiently large order $n$, we characterize all connected graphs satisfying $S_k(G)<2k$. As a consequence, we prove that the path $P_n$ is the unique minimizer of $S_k(G)$ among all connected graphs of order $n$. \vspace*{2mm} We further investigate the first Hoffman-type range \[ 2k\leq S_k(G)<2k+\sqrt{2+\sqrt5}-2. \] We completely characterize the non-tree graphs in this range and reduce the tree case to several explicit families. The proofs combine Ky Fan's variational principle, spectral estimates from vertex-disjoint subgraphs, structural results for graphs with small spectral radius, and long-path arguments for bounded-degree graphs.